Question: Simplify the following expression and state the condition under which the simplification is valid: $r = \dfrac{t^2 + 4t - 5}{t^2 + 8t + 15}$
Explanation: First factor the expressions in the numerator and denominator. $ \dfrac{t^2 + 4t - 5}{t^2 + 8t + 15} = \dfrac{(t - 1)(t + 5)}{(t + 3)(t + 5)} $ Notice that the term $(t + 5)$ appears in both the numerator and denominator. Dividing both the numerator and denominator by $(t + 5)$ gives: $r = \dfrac{t - 1}{t + 3}$ Since we divided by $(t + 5)$, $t \neq -5$. $r = \dfrac{t - 1}{t + 3}; \space t \neq -5$